{
  "id": "1201.2206",
  "version": "v1",
  "published": "2012-01-10T23:26:22.000Z",
  "updated": "2012-01-10T23:26:22.000Z",
  "title": "Dispersive estimates for Schrödinger operators in dimension two with obstructions at zero energy",
  "authors": [
    "M. Burak Erdogan",
    "William R. Green"
  ],
  "comment": "41 pages",
  "journal": "Trans. Amer. Math. Soc. 365 (2013), 6403-6440",
  "doi": "10.1090/S0002-9947-2013-05861-8",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We investigate $L^1(\\R^2)\\to L^\\infty(\\R^2)$ dispersive estimates for the Schr\\\"odinger operator $H=-\\Delta+V$ when there are obstructions, resonances or an eigenvalue, at zero energy. In particular, we show that the existence of an s-wave resonance at zero energy does not destroy the $t^{-1}$ decay rate. We also show that if there is a p-wave resonance or an eigenvalue at zero energy then there is a time dependent operator $F_t$ satisfying $\\|F_t\\|_{L^1\\to L^\\infty} \\lesssim 1$ such that $$\\|e^{itH}P_{ac}-F_t\\|_{L^1\\to L^\\infty} \\lesssim |t|^{-1}, \\text{for} |t|>1.$$ We also establish a weighted dispersive estimate with $t^{-1}$ decay rate in the case when there is an eigenvalue at zero energy but no resonances.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-01-10T23:26:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "zero energy",
      "dispersive estimate",
      "schrödinger operators",
      "obstructions",
      "decay rate"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "AMS",
      "journal": "Trans. Amer. Math. Soc."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1201.2206B"
    }
  }
}